Selection of Terms in an Arithmetic Progression
Sometimes we need to assume certain number of terms in Arithmetic Progression. The following ways are generally used for the selection of terms in an arithmetic progression.
(i) If the sum of three terms in Arithmetic Progression be given, assume the numbers as a - d, a and a + d. Here common difference is d.
(ii) If the sum of four terms in Arithmetic Progression be given, assume the numbers as a - 3d, a - d, a + d and a + 3d.
(iii) If the sum of five terms in Arithmetic Progression be given, assume the numbers as a - 2d, a - d, a, a + d and a + 2d. Here common difference is 2d.
(iv) If the sum of six terms in Arithmetic Progression be given, assume the numbers as a - 5d, a - 3d, a - d, a + d, a + 3d and a + 5d. Here common difference is 2d.
Note: From the
above explanation we understand that in case of an odd number of terms, the
middle term is ‘a’ and the common difference is ‘d’.
Again, in case of an even number of terms the middle terms are a - d, a + d and the common difference is 2d.
Solved examples to observe how to use the selection of terms in an arithmetic progression
1. The sum of three numbers in Arithmetic Progression is 12 and the sum of their square is 56. Find the numbers.
Solution:
Let us assume that the three numbers in Arithmetic Progression be a - d, a and a + d.
According to the problem,
|
Sum = 12 and
⇒ a - d + a + a + d = 12 ⇒ 3a = 12 ⇒ a = 4 |
Sum of the squares = 56
(a - d)\(^{2}\) + a\(^{2}\) + (a + d)\(^{2}\) = 56 ⇒ a\(^{2}\) - 2ad + d\(^{2}\) + a\(^{2}\) + a\(^{2}\) + 2ad + d\(^{2}\) = 56 ⇒ 3a\(^{2}\) + 2d\(^{2}\) = 56 ⇒ 3 × (4)\(^{2}\) + 2d\(^{2}\) = 56 ⇒ 3 × 16 + 2d\(^{2}\) = 56 ⇒ 48 + 2d\(^{2}\) = 56 ⇒ 2d\(^{2}\) = 56 - 48 ⇒ 2d\(^{2}\) = 8 ⇒ d\(^{2}\) = 4 ⇒ d = ± 2 |
If d = 3, the numbers are 4 – 2, 4, 4 + 2 i.e., 2, 4, 6
If d = -3, the numbers are 4 + 2, 4, 4 - 2 i.e., 6, 4, 2
Therefore, the required numbers are 2, 4, 6 or 6, 4, 2.
2. The sum of four numbers in Arithmetic Progression is 20 and the sum of their square is 120. Find the numbers.
Solution:
Let us assume that the four numbers in Arithmetic Progression be a - 3d, a - d, a + d and a + 3d.
According to the problem,
|
Sum = 20
⇒ a - 3d + a - d + a + d + a + 3d = 20 ⇒ 4a = 20 ⇒ a = 5 |
and |
Sum of the squares = 120
⇒ (a - 3d)\(^{2}\) + (a - d)\(^{2}\) + (a + d)\(^{2}\) + (a + 3d)\(^{2}\) = 120 ⇒ a\(^{2}\) - 6ad + 9d\(^{2}\) + a\(^{2}\) - 2ad + d\(^{2}\) + a\(^{2}\) + 2ad + d\(^{2}\) + a\(^{2}\) + 6ad + 9d\(^{2}\) = 120 ⇒ 4a\(^{2}\) + 20d\(^{2}\) = 120 ⇒ 4 × (5)\(^{2}\) + 20d\(^{2}\) = 120 ⇒ 4 × 25 + 20d\(^{2}\) = 120 ⇒ 100 + 20d\(^{2}\) = 120 ⇒ 20d\(^{2}\) = 120 - 100 20d\(^{2}\) = 20 ⇒ d\(^{2}\) = 1 ⇒ d = ± 1 |
If d = 1, the numbers are 5 - 3, 5 - 1, 5 + 1, 5 + 3 i.e., 2, 4, 6, 8
If d = -1, the numbers are 5 + 3, 5 + 1, 5 - 1, 5 - 3 i.e., 8, 6, 4, 2
Therefore, the required numbers are 2, 4, 6, 8 or 8, 6, 4, 2.
3. The sum of three numbers in Arithmetic Progression is -3 and their product is 8. Find the numbers.
Solution:
Let us assume that the three numbers in Arithmetic Progression be a - d, a and a + d.
According to the problem,
|
Sum = -3 and
⇒ a - d + a + a + d = -3 ⇒ 3a = -3 ⇒ a = -1 |
Product = 8
⇒ (a - d) (a) (a + d) = 8 ⇒ (-1)[(-1)\(^{2}\) - d\(^{2}\)] = 8 ⇒ -1(1 - d\(^{2}\)) = 8 ⇒ -1 + d\(^{2}\) = 8 ⇒ d\(^{2}\) = 8 + 1 ⇒ d\(^{2}\) = 9 ⇒ d = ± 3 |
If d = 3, the numbers are -1 - 3, -1, -1 + 3 i.e., -4, -1, 2
If d = -3, the numbers are -1 + 3, -1, -1 - 3 i.e., 2, -1, -4
Therefore, the required numbers are -4, -1, 2 or 2, -1, -4.
● Arithmetic Progression
- Definition of Arithmetic Progression
- General Form of an Arithmetic Progress
- Arithmetic Mean
- Sum of the First n Terms of an Arithmetic Progression
- Sum of the Cubes of First n Natural Numbers
- Sum of First n Natural Numbers
- Sum of the Squares of First n Natural Numbers
- Properties of Arithmetic Progression
- Selection of Terms in an Arithmetic Progression
- Arithmetic Progression Formulae
- Problems on Arithmetic Progression
- Problems on Sum of 'n' Terms of Arithmetic Progression
11 and 12 Grade Math
From Selection of Terms in an Arithmetic Progression to HOME PAGE
Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.
Recent Articles
-
Word Problems on Area and Perimeter | Free Worksheet with Answers
Jul 26, 24 04:58 PM
word problems on area and perimeter -
Worksheet on Perimeter | Perimeter of Squares and Rectangle | Answers
Jul 26, 24 04:37 PM
Practice the questions given in the worksheet on perimeter. The questions are based on finding the perimeter of the triangle, perimeter of the square, perimeter of rectangle and word problems. I. Find… -
Perimeter and Area of Irregular Figures | Solved Example Problems
Jul 26, 24 02:20 PM
Here we will get the ideas how to solve the problems on finding the perimeter and area of irregular figures. The figure PQRSTU is a hexagon. PS is a diagonal and QY, RO, TX and UZ are the respective d… -
Perimeter and Area of Plane Figures | Definition of Perimeter and Area
Jul 26, 24 11:50 AM
A plane figure is made of line segments or arcs of curves in a plane. It is a closed figure if the figure begins and ends at the same point. We are familiar with plane figures like squares, rectangles… -
5th Grade Math Problems | Table of Contents | Worksheets |Free Answers
Jul 26, 24 01:35 AM
In 5th grade math problems you will get all types of examples on different topics along with the solutions. Keeping in mind the mental level of child in Grade 5, every efforts has been made to introdu…
New! Comments
Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.